3.2053 \(\int \frac{1}{(d+e x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=269 \[ \frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \sqrt{e} \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d^2 - a*e^2)*(d + e*x)^(7/2))
+ (5*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d^2 - a*e^2)^2*(d +
e*x)^(5/2)) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*(c*d^2
- a*e^2)^3*(d + e*x)^(3/2)) + (5*c^3*d^3*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(8*Sqrt[e]*(c*d^2 -
a*e^2)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.522784, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{5 c^3 d^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \sqrt{e} \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d^2 - a*e^2)*(d + e*x)^(7/2))
+ (5*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d^2 - a*e^2)^2*(d +
e*x)^(5/2)) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*(c*d^2
- a*e^2)^3*(d + e*x)^(3/2)) + (5*c^3*d^3*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(8*Sqrt[e]*(c*d^2 -
a*e^2)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 104.91, size = 250, normalized size = 0.93 \[ \frac{5 c^{3} d^{3} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{8 \sqrt{e} \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} - \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{5 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

5*c**3*d**3*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d
+ e*x)*sqrt(a*e**2 - c*d**2)))/(8*sqrt(e)*(a*e**2 - c*d**2)**(7/2)) - 5*c**2*d*
*2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*(d + e*x)**(3/2)*(a*e**2 -
c*d**2)**3) + 5*c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(12*(d + e*x)
**(5/2)*(a*e**2 - c*d**2)**2) - sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(
3*(d + e*x)**(7/2)*(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.59994, size = 192, normalized size = 0.71 \[ \frac{\sqrt{d+e x} \left (\frac{(a e+c d x) \left (8 a^2 e^4-2 a c d e^2 (13 d+5 e x)+c^2 d^2 \left (33 d^2+40 d e x+15 e^2 x^2\right )\right )}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac{5 c^3 d^3 \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\sqrt{e} \left (a e^2-c d^2\right )^{7/2}}\right )}{8 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[d + e*x]*(((a*e + c*d*x)*(8*a^2*e^4 - 2*a*c*d*e^2*(13*d + 5*e*x) + c^2*d^2
*(33*d^2 + 40*d*e*x + 15*e^2*x^2)))/(3*(c*d^2 - a*e^2)^3*(d + e*x)^3) + (5*c^3*d
^3*Sqrt[a*e + c*d*x]*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]]
)/(Sqrt[e]*(-(c*d^2) + a*e^2)^(7/2))))/(8*Sqrt[(a*e + c*d*x)*(d + e*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 454, normalized size = 1.7 \[{\frac{1}{24\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{3}{c}^{3}{d}^{3}{e}^{3}+45\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{3}{d}^{4}{e}^{2}+45\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{3}{d}^{5}e+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{3}{d}^{6}-15\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+10\,xacd{e}^{3}\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-40\,x{c}^{2}{d}^{3}e\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{a}^{2}{e}^{4}+26\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}ac{d}^{2}{e}^{2}-33\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a
*e^2-c*d^2)*e)^(1/2))*x^3*c^3*d^3*e^3+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d
^2)*e)^(1/2))*x^2*c^3*d^4*e^2+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(
1/2))*x*c^3*d^5*e+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^
6-15*x^2*c^2*d^2*e^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+10*x*a*c*d*e^3*(c
*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-40*x*c^2*d^3*e*(c*d*x+a*e)^(1/2)*((a*e^2
-c*d^2)*e)^(1/2)-8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4+26*((a*e^2-
c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2-33*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+
a*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/(c*d*x+a*e)^(1/2)/(a*e^2-c*d^2)^3/((a*e^2-c*d^
2)*e)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.23733, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/48*(2*(15*c^2*d^2*e^2*x^2 + 33*c^2*d^4 - 26*a*c*d^2*e^2 + 8*a^2*e^4 + 10*(4*c
^2*d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2
*e + a*e^3)*sqrt(e*x + d) + 15*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*
e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*log(-(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x)*(c*d^2*e - a*e^3)*sqrt(e*x + d) + (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*
d*e^2)*sqrt(-c*d^2*e + a*e^3))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^3*d^10 - 3*a*c^2*
d^8*e^2 + 3*a^2*c*d^6*e^4 - a^3*d^4*e^6 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2
*c*d^2*e^8 - a^3*e^10)*x^4 + 4*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7
- a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 - a^3*d^2*
e^8)*x^2 + 4*(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7)*x)*sq
rt(-c*d^2*e + a*e^3)), 1/24*((15*c^2*d^2*e^2*x^2 + 33*c^2*d^4 - 26*a*c*d^2*e^2 +
8*a^2*e^4 + 10*(4*c^2*d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d) - 15*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e
^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*arctan(sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*
e^2 + (c*d^2*e + a*e^3)*x)))/((c^3*d^10 - 3*a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4 - a^
3*d^4*e^6 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10)*x^4 + 4
*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^3 + 6*(c^3*d^8*
e^2 - 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e - 3*a*
c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7)*x)*sqrt(c*d^2*e - a*e^3))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.579636, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

sage0*x